New Proofs of Some Dedekind $\eta$-Function Identities of Level $6$

Abstract

Recently, Shaun Cooper proved several interesting
$\eta$-function identities of level $6$ while finding series
and iterations for ${1}/{\pi}.$ In this sequel,
we present some new proofs of the $\eta$-function identities
of level $6$ discovered by Cooper. Here, in this article,
we make use of the modular equation of degree $3$ in
two methods. We further give some
interesting combinatorial interpretations
of colored partitions.